Reload Pattern

8/14/2019 Reload Pattern

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Development of parallellized higher-ordergeneralized depletion perturbation theory forapplication in equilibrium cycle optimization

R. van Geemert *, J.E. HoogenboomInterfaculty Reactor Institute, Mekelweg 15, 2629 JB, Delft, The Netherlands

Received 25 October 2000; accepted 20 November 2000

Abstract

As nuclear fuel economy is basically a multi-cycle issue, a fair way of evaluating reload

patterns is to consider their performance in the case of an equilibrium cycle . The equilibriumcycle associated with a reload pattern is dened as the limit fuel cycle that eventually emergesafter multiple successive periodic refueling, each time implementing the same reload scheme.Since the equilibrium cycle is the solution of a reload operation invariance equation, it can inprinciple be found with su cient accuracy only by applying an iterative procedure, simulatingthe emergence of the limit cycle. For a design purpose such as the optimization of reloadpatterns, in which many di erent equilibrium cycle perturbations (resulting from many di er-ent limited changes in the reload operator) must be evaluated, this requires far too much com-putational e ort. However, for very fast calculation of these many di erent equilibrium cycleperturbations it is also possible to set up a generalized variational approach. This approachresults in an iterative scheme that yields the exact perturbation in the equilibrium cycle solution

as well, in an accelerated way. Furthermore, both the solution of the adjoint equations occur-ring in the perturbation theory formalism and the implementation of the optimization algo-rithm have been parallellized and executed on a massively parallel machine. The combinationof parallellism and generalized perturbation theory o ers the opportunity to perform veryexhaustive, fast and accurate sampling of the solution space for the equilibrium cycle reloadpattern optimization problem. # 2001 Elsevier Science Ltd. All rights reserved.

Annals of Nuclear Energy 28 (2001) 13771411

www.elsevier.com/locate/anucene

0306-4549/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved.P I I : S0306 -4549 (00 )00135 -3

* Corresponding author at current working address: Paul Scherrer Institute, CH-5232 Villigen PSI,Switzerland.

E-mail addresses: [email protected] (R. van Geemert), [email protected] (J.E. Hoo-genboom).


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1. Introduction

In order to evaluate the `quality' of a reloading scheme, it is important not tostudy the associated fuel economy for only the forthcoming cycle. Instead, somemethod should be available to gain some indication about the scheme's multi-cycleperformance. The most obvious and interesting multi-cycle evaluational method isto study the equilibrium cycle behaviour. The equilibrium cycle associated with areload pattern is dened as the limit fuel cycle that eventually emerges after multiplesuccessive periodic refueling with the same reload pattern. In reload pattern opti-mization, search algorithms are mostly based on the assessment of the e ects of permutations in the fuel bundle shu ing scheme, followed by acceptance decisionsin which the permutation yielding the largest improvement in the objective functionvalue (while satisfying the operational and safety constraints) is chosen to determinethe next reference pattern in the search procedure. Generally, the exact value of theobjective function for an equilibrium cycle can only be determined by implementinga forward iterative procedure to obtain the equilibrium cycle solution from BOC toEOC. This implies that reload pattern optimization methods in which large numbersof di erent refueling schemes must be evaluated in this way are quite expensive from acomputational point of view. Hence it would be helpful to have accelerated iterativemethods available for fast assessment of di erent reloading schemes in such a waythat the exact perturbed equilibrium cycle characteristics can be obtained with con-siderably reduced computational e ort. A lot of work has already been done in the

eld of applying perturbation theory to in-core fuel management optimization, butmost of it is dedicated to non-equilibrium cycle fuel management (White and Avila,1990; Maldonado et al., 1995; Moore and Turinsky, 1998; Van Geemert et al., 1998b).

Yang and Downar (1989) did develop perturbation theoretical methods for theequilibrium cycle, in order to assess the e ects of changes in general reactor physicsparameters (such as material properties or the enrichment level of fresh fuel assem-blies) on the required feed enrichment or on the cycle length for a constrained equili-brium cycle (with the reload operator xed). Constrained here means that either thefeed enrichment or the cycle length are dictated by the condition that the e ectivemultiplication factor ke ( u c)(EOC) of the uncontrolled core (without external reactiv-

ity control, such as control rods or soluble boron) at EOC is xed at a certain value(usually unity). In their paper, either the feed enrichment is xed, whereas the lengthof the equilibrium cycle is subject to this EOC condition, or vice versa. Unfortunately,Yang and Downar only reported results for zero-dimensional cases (so with a uni-form ux in the entire reactor core) and for a xed reload pattern. However, forapplication in loading pattern optimization, it is of course necessary to account forspatial variations in the ux distribution and to consider changes in a far moreabstract, discrete equilibrium cycle parameter, which is the reload pattern itself . Instudying equilibrium cycle behaviour, one should realize that a change in the shuf-ing scheme a ects the entire BOC fuel density distribution. The generalized per-

turbation theoretical approach for the equilibrium cycle proposed by us will beshown to result in an accelerated iterative scheme that yields the exact perturbationin the equilibrium cycle solution (caused by a limited change in the reload operator)

1378 R. van Geemert, J.E. Hoogenboom/ Annals of Nuclear Energy 28 (2001) 13771411


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in considerably fewer iterative steps and thus with a signicantly reduced computa-tional e ort.

2. General aspects of equilibrium cycle optimization

As is generally known (Stevens, 1995, Van Geemert, 1999), no genuine orderingprinciple exists for the reload patterns or, in other words, it is not possible to mapthe solution space into regions (of comparable performances) which can be easilydened in terms of the variables of the reload operator. However, it seems that it ispossible for the equilibrium cycle to establish some rough link between the shape of the power distribution and the equilibrium cycle fuel economy. Utilizing a simpliedbut rather elegant model (De Jong, 1995), it can be shown that equilibrium cycles aregenerally more economical when they are characterized by power distribution shapesthat are at with regard to both space and time. Via Haling's theorem (stating that anoperation cycle's total maximal power peaking factor during the cycle is minimal inthe case of a constant power distribution shape throughout the cycle) this generalcorrelation can be translated into the following rough property:loading schemes thatare economical in terms of the equilibrium cycle generally give rise to rather constantand at power distributions which thus yield low maximal power peaking factors.

Obviously, knowledge of the existence of this general, rough correlation does notreduce the complexity of the loading pattern optimization problem in any way not

only because it is only a rough correlation but also since it is impossible to establishan easy mathematical relationship between the way in which the core is conguredand the resulting time-dependent power distributions. Two extreme examples of loading pattern types are the centre-to-outside loading (COL) and the outside-to-center loading (OCL). In the case of COL loading, the fresh fuel assemblies areplaced in the centre of the core, whereas the burnt fuel assemblies will be placedcloser to the periphery as their burnup increases. In the case of OCL loading thereverse happens. Generally, in the equilibrium situation they will both give rise tonon-optimal operation cycle behaviour with an additional unacceptably high powerpeaking factor in the COL case. In practise, power distributions that are at and

that change minimally throughout the operation cycle are to be achieved by, forexample, checkerboard-like patterns with ring structures in the placement of thedi erent fuel ages. And hence in general, loading pattern design can be done quitesatisfactorily by the application of rough engineering knowledge translated intosome general rules of thumb for where and where not to place fuel assemblies withdi erent histories. But very obviously, a huge freedom still exists in the design of such patterns. Further optimization starting from a pattern obtained in this way canonly be achieved by means of implementing automated optimization proceduresfeaturing inherent mathematical, physical or even biological mechanisms as the driv-ing force of their e ectiveness. Well-known examples of these are cyclic interchange

optimization methods (De Jong, 1995; Van Geemert, 1999), simulated annealingmethods (Kropaczek, 1991; Parks, 1987; Verhagen, 1993; Smuc, 1994; Stevens, 1995;Van Geemert, 1998a) and evolutionary optimization algorithms (DeChaine 1996;

R. van Geemert, J.E. Hoogenboom/ Annals of Nuclear Energy 28 (2001) 13771411 1379


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Parks, 1996; Axmann, 1997; Van Geemert, 1999). Since in principle, the equilibriumcycle can only be determined exactly as the converged outcome of a computationallyexpensive calculational procedure, these optimization algorithms can be quite time-consuming if no attempts are made to accelerate the iterative calculations. Forequilibrium cycle optimization, it is worthwile to implement perturbation theoryapproaches at two di erent levels:

(i) Acceleration of the iterative solutions of the quasi-static neutronics equationsby application of generalized perturbation theory (GPT).

(ii) Acceleration of the perturbed limit cycle iterations by application of an extendedtime-dependent generalized variational approach, which we call generalizedequilibrium cycle depletion perturbation theory (GECDPT). In this approach,the classical depletion perturbation theory (Williams, 1979) is embedded.

The GPT approach associated with acceleration technique (i), which is not reallyspecic for equilibrium cycle optimization purposes, has already been applied suc-cesfully for non-equilibrium cycle optimization purposes (Maldonado et al., 1995;Moore and Turinsky, 1997, Van Geemert et al., 1998b). The most important line of research adopted in this work was to investigate whether a similar higher-orderiterative method, to be referred to as GECDPT, could be set up for the equilibriumcycle. In the single time-step GPT, a generalized response functional is dened thatconsists of a response funtion and a summation of constraint functions formed byinner products of Langrange multipliers (later to be referred to as the adjoint elds )

and the system equations that govern the time evolution of the reactor. This is donein such a way that the constraint functions vanish when the system equations areexactly satised. GECDPT involves the set-up of an extended generalized responsefunctional that includes the cyclic reload operation invariance equation as a con-straint function in addition to the other, more traditional constraint functions (i.e.the ux shape l -eigenvalue equation, the power normalisation equation, the deple-tion equations, etc.). As it turns out, an adjoint cyclic invariance equation for theadjoint time-dependent nuclide density eld surfaces that, if satised, will give rise tothe disappearance of the principal term in the development of the generalized func-tional change induced by a discrete change in the reload operator. This term will be

shown to be dominant with respect to the higher-order terms for su ciently modestchanges in the reload operator, as a result of which the higher-order iterative schemewill quickly converge to the exact perturbed equilibrium cycle. The GECDPTapproach follows from the analysis of an extended form of the GPT functional.Because of this, and because the single time-step GPT method has been applied inthis study, we will now provide an introduction into the single time-step GPTmethod, so as to facilitate the introduction of the GECDPT formalism.

3. The single time-step GPT method

The time evolution of a PWR core is basically described by the energy-, space-,and time-dependent neutron ux t and by the space- and time-dependent nuclide



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density matrix x t . Generally, since the neutronic and depletion equations are cou-pled, a quasi-static approach, featuring a number of time steps t i , must be adopted fornumerical simulation of how the core evolves from BOC to EOC. In this approach, itis assumed that the neutronics eigenvalue equation, determining the spatial ux shape

i , can be written compactly as

L i l iF i i 0 I

where the eigenvalue l is the reciprocal of the e ective multiplication factor of theuncontrolled (=without external reactivity control) core ke and L and F are the lossand production operator, respectively. The normalisation requirement for the spatialux shape vector can be written as

ll nodes

1 P

with the index I denoting the di erent core nodes in the system geometry. In the 1 12-group approximation adopted in this study, there is e ectively only one energy group,but the general GPT and GECDPT formulations presented here can be easily exten-ded for incorporation of multiple energy groups. When determining the responsefunctional, one should explicitly account for the fact that a change in the nuclidedensity eld also inuences the neutron ux eld. In conformity with the variational

approach, this can be realized by treating the neutronics and ux normalisationequations as constraints on the response quantity itself, and appending them to theresponse function using Lagrange multipliers. Many inner product denitions occur,which can be most conveniently written using the Dirac bracket notations:

j ; QFor the eigenvector and the eigenvalue l , dened at a certain time step t i , two

coupled functionals can be dened, in which the quasi-static equations are appendedto the response functions using the Lagrange multipliers , a

"and . For the sake

of simplicity, we now abandon the time index (i)

1 j L l F h i a 1 j h i 1 2 l

j L l F h i j F h i

VbbbbbbbbX

R

We note that, due to the commutativity property of the adjoint operators, we canwrite:

j L l F h i L l F j h i S



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with L and F the adjoint operators of L and F (occurring in the denition of 2,respectively. The same applies to the inner product j L l F , which can bewritten as L l F j . If l and are exact solutions to the quasi-static equa-tions, then 1 and 2 l irrespective of the choices for , a and . If thenuclide density distribution x is perturbed ( x 3 x H x x ), 1 and 2 willbe perturbed as well, since the change in x e ects a change in the operators L and F :

1;2 3 H1;2 LH; FH; H; l H T

where the prime variables refer to their perturbed values. Again, if l H and H areexact solutions to the perturbed equations, then H1

Hand H2 lH. In single-time

step GPT, the Lagrange multipliers , a and are numerically conditioned in sucha way that the following iterative system,

m 1 m j L H l m FH m h i a 1 j m h i 1

l m 1 l m j L H l m FH

m h i j FH m h i

;VbbbbbbbbX

U

if convergent (depending on how much L Hand FHdi er from L and F ), converges tothe exact perturbed solution ( l H; H) in a number of steps that becomes very small for

small changes L and F . The conditioning of

, a and

follows from a numberof Euler stationarity conditions which for the specic rst-order components in thedevelopment for 1 and 2 that contain perturbations in forward quantities thatare still to be determined ( l and ), to vanish. Substracting the expressions for theperturbed and the unperturbed response functional and ordering the terms gives thedevelopments

m 1 m j L l F h il m j F h i L l F 1 j m

h i1st order

j L l F m h il m j F h i2nd order l m j F m h i3rd order

V

and

l m 1 L l F j m h i j L l F m h i

j FH m

h iW

Noting that m m can be written as 1 j m with 1 the unity matrix,Eq. (8) can be written for this case as:



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m 1 j L l F h il m j F h i L l F 1 1 j m

h i1st order

j L l F h i2nd order l m j F m h i3rd order

IH

The third and the second term on the right will vanish if the adjoint ux shapeequation

L l F 1 1 II

and the orthogonality condition

j F h i0 IPare satised. For argueing how this combined objective can be realized, it isworthwile to consider the eigenset l n ;

n Y n 0; 1; 2; F F Fn oof the adjoint ux shapel -eigenvalue equation:

L l n F

n 0 IQ

which is, of course, fully analogous to the standard eigenset l n ; n Y n 0; 1; 2; F F Fn o, of the normal ux shape l -eigenvalue equation:L l n F n 0 IR

Generally, l n l n . The fundamental adjoint ux shape 0 (conventionally writ-

ten simply as , like 0 is written as ) is a very important quantity. The basicreason for this is the validity of a mathematical orthogonality property that can beproved in the following way: after premultiplication of Eq. (14) (and abandoning the

time index i) with the l th eigenvector l , recalling the commutativity property

l j

L n L l j n and employing Eq. (13), we obtain

l l l j F nh il n l j F nh i IS

Hence, since l n T l l in the case n T l , the inner product l j F n should vanishfor the case n T l . Thus, the validity of the property

l j F nh icn nl IThas been established. The implication of this property is that, if some adjoint uxshape is to be orthogonal to the ssion rate distribution F , it should not containany component of i.e. it should be orthogonal to . So, Eq. (12) basically



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requires that the Lagrange multiplier is not allowed to contain any component of the fundamental solution of the homogeneous adjoint equation L l F 0.This fundamental mode is the only eigenstate which does not yield zero when formedthe inner product with F , since the eigensets and

nYn 0; 1; 2; F F Fand nYn0; 1; 2; F F Fobey the general orthogonality property l j F n cn nl . Thus,

n j F

unequals zero only in the case n=0. More specically, if p is the particular solution toEq. (11) satisfying Eq. (12) such that p j F vanishes, where is the fundamentalsolution to the homogeneous equation, then p b

is also a solution of Eq. (11) forall b. However, due to Eq. (12) the value of b is xed to be zero, so that will not be`contaminated' with the fundamental solution . In each step in the iterative process of solving Eq. (11), this can be e ected by application of the ltering operation

X j F

h i j F h i IU

The choice for the multiplier a is determined by the fact that Eq. (14) and Eq. (11)specify that j Q 0. This becomes obvious when analysing the inner product

j Q :

j Q h i j L l F j h i IVL

lF j

h i0 j

h i0 IWHence, is given by1 j h i1 j h i

PH

because of the normalisation condition (2) on . If Eqs. (11), (12) and (20) aresatised by and a, the expression for the rst-order prediction of the change in the

response ux , denoted by1

, is1 j L l F h i PINeedless to say, this expression yields an accurate result only in case of very small

changes L and F . Regarding Eq. (9) since L l F 0, the rst term in thenominator of (9) vanishes, and thus we obtain:

m 1 l j L l F m h i

j FH m

h iPP

With the nodal ux distribution as the response vector for functional 1 and thel -eigenvalue as the scalar response quantity for functional 2, these expressions



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eventually yield a higher-order iterative scheme from which the exact perturbednodal ux distribution can be obtained at low computational cost:

m 1 1 j L m l F m

h il j F m h i

m 1 l j L l F m h i

j FH m h iVbbbbbbbbX

PQ

This scheme has the convenient property that it will converge very rapidly whenthe perturbation operators L and F are relatively small with respect to L and F .This can be made clear by pointing out that from Eq. (11) it follows that

% L l F 1 1L l F

PR

so that the condition number for the scheme (23) becomes

%L l F L l F

PS

Hence, the smaller L and F , the faster the convergence. This will generally be the

case for an appropriately conditioned numerical design strategy such as reload pat-tern design optimization, in which the results of many di erent small changes in thesystem need to be evaluated during the optimization procedure. We stress that onlywhen many of such changes are to be evaluated, the evaluational economy will out-weigh the extra computational e ort involved in determining the adjoint elds. Itshould be pointed out here that scheme (23) results from the exact response functionalexpansion by explicit implementation of the supposedly exact validity of the adjointEqs. (11) and (20). Hence if this validity is limited because of a limited number of iterative steps available for solving Eqs. (11) and (20), small errors can be expected.This problem vanishes when one applies the iterative scheme (7) that is based on the

original exact response functional expansion, without ommitting any terms. Sincethe same adjoint elds and a are used as in scheme (23), the terms that areneglected in scheme (7) will become very small all right, such that the same con-vergence behaviour will emerge. Thus, scheme (7) is the scheme to implement, andthe signicance of scheme Eq. (23) reduces to a mere theoretical one, since it reectsthe inner dynamics of scheme (7) explicitly, and claries why fast convergence occursfor small changes in L and F caused by a limited change X in the reload operator.

4. Generalized equilibrium cycle depletion perturbation theory

Since the time dependent neutronics and nuclide density elds implicitly dependon one another, a spatial perturbation in the nuclide density eld at BOC (so at



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t t0 ) will perturb the entire neutron/nuclide eld for the forthcoming cycle. Inclassical depletion perturbation theory (DPT) (Gandini, 1975; Williams, 1979), it isconventionally assumed that the perturbed BOC nuclide density distribution isknown exactly. Due to the fact that the neutronics eld and the nuclide density eldare governed by a set of coupled (di erential) equations and cannot be varied inde-pendently, it is impossible to calculate directly the exact way in which a perturbationin the combined neutron/nuclide eld will propagate from BOC to EOC. Any datachange that changes one eld will also change the other eld, because the two eldsare constrained to ``move'' only in a fashion consistent with their coupled eldequations. This will be even more complicated when one is dealing with an equili-brium cycle, in which case one wants to evaluate the e ect of a change in thereload operator on the equilibrium cycle solution. The reload operator can berepresented mathematically by a square binary matrix X, the elements of which areto be dened by

1 if the fuel element th t h s resided in node is lo ted in node fter relo ding

0 otherwiseVX

PT

If 0 then obviously the fuel element that has resided in node J is to bedischarged. If 0 then apparently none of the older fuel elements which

were already present in the core will be place in node I during reloading. In thatcase, node I is to receive a fresh fuel element. If the fuel elements are characterizedby their nuclide density vectors, the invariance of the equilibrium cycle with respectto the reload operation can be mathematically dened as:

x t 0 Xx tE

1 X x p PUx p is a convenient notation for x p ; x p ; F F F; x p with x p the vector denoting

the densities of the di erent nuclides in the fresh fuel. The symbol 1 indicates the

unity operator. The equilibrium cycle can be determined iteratively using thisinvariance condition. Starting from an initial guess for the equilibrium cycle, suc-cessive cycles are simulated, with at each reloading the application of the samereload scheme and the same composition of the fresh fuel assemblies, until a con-verged limit cycle emerges. For compact representation and notation of equibriumcycle reload operators, one can apply the fact that reload schemes for N-node sys-tems, featuring n age groups for the fuel elements (batches), lead to N/n fuel bundlelife history trajectories. These trajectories represent the sequences of fuel positionsthat act as hosts for a particular fuel element during its time in the reactor core(normally 4 cycles, so n=4), as it travels from one position to the next until it is

discharged after n cycles. The collection of all such history trajectories is a con-venient and compact representation of the associated reload pattern, and can bewritten as



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H

i 1 1 3 i1

2 3 F F F 3 i1

nF F F F F F F F F F F FF F F F F F F F F F F FF F F F F F F F F F F FF F F F F F F F F F F Fi

xn

1 3 ixn

2 3 F F F 3 ixn

n

VbbbbbbbbbbbbX

WbbbbbbabbbbbbY

PV

In this denotation, in which for example the rst row of H denotes the rst fuelelement life history trajectory, each arrow represents an individual fuel elementrepositioning operation. The nonzero elements in the matrix X follow from

i j m ; i j m 1 1 for ll m 1; F F F n 1; j 1; F F F;xn : PW

In order to account for the e ect on the equilibrium cycle of a change in the reloadoperator, an extra term representing the reload operation invariance condition needsto be added to the response functional. Needless to say, this will make the determi-nation of the properly chosen adjoint nuclide density eld somewhat more compli-cated. Naturally, in the more general time-dependent picture a number of additionalconstraint functions must be included in the functional, corresponding to thedepletion processes and the refueling operation which occurs periodically in time.

Again, all the system equations can be treated as constraints on the response, andappended to the response function using appropriate Lagrange multipliers. Since thereload operation takes place in the nuclide density space, the reload equation willhave the same Lagrange multiplier as the nuclide transmutation equation. First of all, we note that the transmutation equation dictating the time evolution of thenuclide density eld can be written as

@@t

x t t x t QHwith denoting the absorption cross sections matrix and D the decay operator. Weemphasize here that the operators active in the nuclide type space are marked with ahat, and that an implicit product in the nuclide type space occurs when the operators

and D appear in combination with the nuclide density eld x t . The fast ux t can be written as a product of the time dependent shape function r ; t (which isnormalized such that r ; t d 1 for all t) and a time dependent power nor-malisation factor t . This allows for a convenient implementation of the conditionthat the reactor power is restricted to remain at a constant level. Using the innerproduct notation, this condition can be written as:

wf i p ; i j ih i tot l for ll t QI



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The response quantity we are interested in is the full collection of all nuclide densitiesat the end of the operation cycle, so dened at t tE. In order to stress that the responseis dened at t tE, we denote the response quantity as x t E

. Applying the Lagrange

multipliers C i , P i and N t , the general response functional X; x t ; x p ; i; i; l ican be written as:x t

1

i 0 t i 1

t iN t j i i

@@t

j x t ( )dt

1

i 0

i j L i l iF i ih i1

i 0

P i wf i p ; i j ih i tot l

t

tN t j

@

@tx t X 1 x t x

p

t t

( )dt

QP

in which the integrand in the last term is the di erential equation describing thereload operation at EOC. Forward integration of this equation from t yields thereload equation x t 0 X Xx t

1 X x p . Now, for the sake of convenience,we think of as being composed of two parts 1 and 2: = 1+ 2. 1 isassociated with the `in-cycle' system equations (describing the `in-cycle' neutronics,power control and burnup) and 2 is associated with the discrete time event of reloading (with respect to which the equilibrium cycle should be invariant):

1

1

i 0 t i 1

t iN t j i i

@@t

j x t ( )dt 1

i 0

C i j L i l iF i ih i

1

i 0

i wf i p ; i j ih i tot l QQ

and

2 x t

t

tN t j

@

@tx t X 1 x t

x p

t t

( )dt QR

If x t , i, l i and i are exact equilibrium cycle solutions to the quasi-staticburnup equations, then x t . Now, if the reload operator X and the fresh fuelcomposition x p (and hence the entire equilibrium cycle solution) are perturbed, thiswill inuence :

3 H XH; x H t ; x Hp ; Hi;

Hi; l

Hi ; QS

where the prime variables refer to their perturbed values. Again, if x H t , Hi, Hi and

l Hi exactly satisfy the perturbed equilibrium cycle system equations dictated by XHand x Hp , then

H x H t , so the perturbed functional value will be equal to theexact perturbed EOC equilibrium cycle nuclide density distribution. Expanding



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about the unperturbed state and neglecting the second-order terms in the continuousvariables, we obtain the rst order prediction for , denoted as 1 :

1 1

i 0 t i 1

t i

@ 1@x t

j x t B Cdt @@l i l i @ 1@ i i @ 1@ i j iB C2 32 X; x p ; x t QT

with

2 X; x p ; x t x t

t

t x

t j@@t x t X x t X x t x p X x t t t (X x t t t 1 XH x p t t idt

QU

We note that 2 X; x p ; x t as dened in Eq. (37) is not a rst orderapproximative expression, but an exact expansion of the change of 2 due to thesimultaneous changes X, x p and x t . This is why no superscript is added. Manyof the contributions to 1 can be forced to vanish by dening the EulerLagrangestationarity equations for the adjoint elds C i , P

i and N

t . For example, the

functional derivative with respect to i is:@ 1@ i

t i 1

t iN t j i j x t h idt P i wf p ;i j ih i QV

For this expression to vanish, the multiplier P i should be chosen such that:

P i1

wf p ; i j ih i t i 1

t iN t j ijx t h idt QW

Employing the commutativity property of the adjoint operators L* and F*, thefunctional derivative with respect to i can be written as

@ 1@ i

t i 1

t iN t j ijx t h idt L i l iF i C

i wf P

i ip ; i RH

In analogy with Eq. (11), this partial derivative will vanish if C i satises theinhomogeneous adjoint ux shape equation

Li l iF

i

C

i Q

i RI

with the adjoint source



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Q i t i 1

t ix t j ijx t h idt wf P i ip ; i RP

The requirement that @ 1@l i vanishes results in the same orthogonality condition asEq. (12), forcing C i to be orthogonal to the neutron production term F i i at t t i:

C i j F i ih i0 RQThe stationarity condition corresponding to a variation in x t is more complex

than for the other variables. This is basically because x t is the only `forward'quantity that is allowed to vary continuously in time. Due to this, a signicant partof its contribution in the variation functional is embedded in an integral, while theother part merely consists of terms dened at discrete equidistant moments t i con-taining the `snapshots' x t i of the nuclide density elds. To derive the Euler condi-tions for N t , we rst consider the variation of 1 with respect to x t :

1 1 x t

1

i 0 t i 1

t iN t j i i

@@t

j x t ( )dt

1

i 0

C i j@L i@x i

l i@F i@x i

j iB CP i wf i @p ; i@x i j iB C4 5: x iRR

Using the adjoint operators

and D

and partial integration with respect to t yields

t i 1

t ix t j i i

@@t

j x t ( )dt N t i j x t i h i N ti 1 j x t

i 1 h i

t i 1

t i i i

@@t N t j x t ( )dt

RS

If we dene

zi C i j @L i@x i l i @F i@x i

j iB CPi wf i @p

;i@x i

j iB C; RTwe can write 1 1 x t as

1 x t T 1

i 0 t i 1

t i i i D

@@t x t jN t ( )dt

x

t i zij N t i h i N

t

i 1

j x t

i 1

h iRU

If we have the N t satisfy the in-cycle time boundary discontinuity conditions



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N ti N t i zi; i 0; 1; F F F; 2; RV

the summation will reduce to

1 1 x t N t0 z0j x t 0 h i N

t j x t

h i1i 0

t i 1

t i i i

@@t N t jx t ( )dt

RW

Further, if the adjoint transmutation equation,

@@t

N t i i

N t ; t i < t < t i 1 SH

is satised, the summation eventually reduces to

1 1 x t N t0 z0j x t 0 h i N

t j x t

h i SIBefore we consider 2x t , impose the stationarity condition for this case andderive the adjoint cyclic boundary condition, we point out some continuity and dis-

continuity properties and conventions for the forward and adjoint nuclide density

elds. For the `in-cycle' intervals characterized by t i < t < t i 1 , the forward time-dependent nuclide density distribution is absolutely continuous in the sense thatx t

i x t i . The only discontinuity for the forward eld occurs at t t , wherethe reload operation is applied, resulting in the reloaded state at t t0 . The adjoint

time-dependent nuclide density distribution however has a discontinuity betweeneach in-cycle interval and the next as well, in accordance with (48). We stress that,strictly mathematically, t t and t t0 coincide due to which, in the framework of the formalism presented here, t t and t t0 are to be thought of as t0 t t ort t t0 . Now, for the adjoint nuclide density distribution we adopt the conventionthat N t i N t i

. Likewise, at the cyclic boundary (dened equivalently by t t0

and t t ), N

t0 N

t0 . For the forward nuclide density distribution we imple-ment x t x t at the cyclic boundary, which is of course adjoint to the deni-tion N t0 N t0 for the adjoint eld and in analogy with the opposed timedirections of forward and backward depletion. These conventions are inspired bythe fact that forward/backward depletion during a time interval yields certain valuesfor the nal/initial nuclide densities for that interval. An important consequence of these conventions occurs in time integrals over the cyclic bounday, involving innerproducts of adjoint and forward elds. For example:

t

tN t j t t x t h idt N

t0 j x t

h i SP



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Utilizing these properties, we will now consider 2 from Eq. (35) and derive theadjoint cyclic boundary condition corresponding to rst-order stationarity of 2with respect to a perturbation in the nuclide density distribution. Realizing thatx t can be written as

x t x t t

t1 t t jx t h idt ; SQ

applying partial integration with respect to t again and using the adjoint reloadmatrix X, 2 can be written as

2 X; x p ; x t

N t0

z0 j x t 0

h iN t

j x t

h i t

t

@@t

X 1 t t N t 1 t t jx t ( )dt

t

tN t j X x H t x Hp 1 X x pn ot t h idt

SR

We note that, in correspondence with Eq. (52),

t

tN t j X x H t x Hp

1 X x p

n ot t

h idt

N t0 j X xH t x

Hp 1 X x ph i; SS

and recall that, for the equilibrium case, x t 0 is related to X, x p and x t

asx t 0 X x t

x p XH x t x p x p ST

Implementing this, and realizing that the rst term in Eq. (54) can be placed insidethe integral constituting the third term, we obtain

1 N t0 z0 j X xH t x

Hp 1 X x ph ii SU

t

t

@N t @t

X 1 N t z0 1 t t jx t ( )dt SVNow, the integral in Eq. (58) will vanish if its integrand vanishes:

@@t

N t X N t z0 N t 1 t t SW



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Integration of this di erential equation from t to t yields

N t

N t

X N t

z0

N t

1 TH

For the equilibrium cycle, t is equivalent to t0 . Thus, the condition that theintegral part of 2 x t disappears results in the following adjoint cyclic bound-ary condition to be satised by the adjoint time-dependent nuclide density distribu-tion N t :

N t X N t0 z0 1 TI

If this condition is satised, 1 can be written very compactly as

1 S0 j X x t xHp 1 X x ph iS0 j X x t

h i TPwith the sensitivity operator S 0 dened as

S0 N t0 z0 TQWe stress that the principal part of this rst-order prediction 1 of the per-

turbed equilibrium cycle EOC eld,

S0 j X x t

xHp

1 X xp

h i TRcan be obtained without any new iterative calculations. Further, it is interesting topoint out the physical interpretation of Eq. (61). This physical interpretation is set ina reversed time picture in which importances are propagated from one fuel elementposition preceding the other in correspondence with an adjoint reload operation.This interpretation is illustrated in Fig. 1. Whereas the normal reload scheme indi-cates in a forward way how disturbances in fresh fuel elements travel through thecore via the periodic shu ing process, the adjoint reload scheme indicates (in abackward way) the history of an observed disturbance in a discharged fuel element,

thus eventually pinpointing from where and when the disturbance originates.Mathematically this implies that the adjoint trajectory representation H * can beobtained from H simply by reversing the orientation of the arrows in expression(28), after which the non-zero elements in X can be obtained with formula (29).

5. Accelerating the limit cycle iterations

The general numerical goal dened for this study is to develop and implement anaccelerated way to perform the equilibrium cycle iteration. In the normal picture this

iteration is done by plain simulation of how a limit cycle is reached in real life, i.e.one simply performs the neutronics and depletion calculations during each mth iter-ated operation cycle, followed by a reloading, each time with the same reload pattern



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of which one wants to obtain the associated equilibrium cycle. This process endswhen the limit cycle has been obtained with su cient accuracy.

In the numerical picture proposed here, we assume that one forward equilibriumcycle solution has been obtained for a specic reference reload pattern. The refer-ence reload operator and its associated equilibrium cycle solution are assumed tohave been adopted as the case for which the adjoint Euler stationarity Eqs. (41) and

(50) have been solved. Further, it is assumed that the adjoint eldsC

and N

t havebeen solved such that the additional conditions (43), (48) and (61) are satised. Now,all the new reload operators of which the associated equilibrium cycle solutions needto be evaluated are treated as perturbations on the reference reload operator.

In the way to perform the perturbed equilibrium cycle iteration proposed here, asignicant calculational acceleration is realized in three ways:

. The iteration is started with the estimate (64) as the 0th iterand, which is therst-order prediction of how the equilibrium cycle EOC nuclide density dis-tribution is perturbed. Via Eq. (56), this can be translated into a prediction of

how the equilibrium cycle BOC nuclide density distribution is perturbed.. The individual `in-cycle' neutronics equations, for determination of the eigen-

values l i and the ux distributions i, i 0; F F F; 1 are performed by sol-ving the iterative scheme (7) instead of applying the power method to Eq. (1).This yields a signicant reduction in the number of iterative steps required toobtain a certain prespecied accuracy level. Since the depletion calculations arecomputationally cheap, these are executed in a normal way ; in this manner,the perturbed time-dependent nuclide densities x Hm t1 ; x Hm t2 ; F F F; x Hm t corresponding with the mth equilibrium cycle iterative step, are obtained.

. After the (T-1)th neutronics calculation has been completed, and x Hm t hasbeen calculated, a correction formula, in which N t is used, is applied inorder to obtain the corrected iterand m 1 orr x t

, before the normal numericalreload operation is performed.

Fig. 1. Forward and adjoint reload operations.



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We will now derive the accelerative expression to be used. First of all, we recall thegeneral shape of the response functional,

x t 1

i 0 t

i 1

t iN t j i i @@t

jx t ( )dt

1

i 0

C i jL i l iF ij ih i1

i 0

P i wf i p ;ij ih i tot l

t

tN t j

@@t

x t X 1 x t x p t t ( )dt TSsuch that the accelerative expression to be implemented is:

m orr x t

m x t

1

i 0 t i 1

t iN t j m i i

@@t x t !( )dt

1

i 0

C i jm L i l iF i i h i

1

i 0

P i wf m i p ; ij ih i

t

tN t j

@@t

X 1 t t m x t X x t x p (t t X m x t t t 1 XH x p t t

idt TT

Since in each mth iterative step the exact l i, i are obtained from the `in-cycle' neu-tronics eigenvalue equations [in an accelerated way through solution of system (7)], allthe individual terms occurring in the summations from i=0 to i=T 1 will vanish:

m i i @@t x t !0;

m L i l iF i ; i ; 0m i p ;i j ih i 0

VbbbbbbbbX

TU

Further, recalling Eqs. (54) and (55), the expression reduces to

m orr x t

m x t N

t jm x t h i N

t0 jm x t 0 h i

t

t

@@t

X 1 t t N t j m x t ( )dtN t0

j X x t

x p

h iN t0

j X m x t

h iN t0 j 1 XH x ph i TV



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We now take into account that, if Eq. (59) is satised, we have

@

@tX 1 t t

!N t 1 t t Xz

0t t ; TW

due to which we obtain

m orr x t

N t j

m x t h i N t0 j

m x t 0 h iXz0 j m x t h i

N t0 j X x t

x p h iN t0 j

m x t r h iN t0

j 1 XH x p

h iUH

Now, we wish to end up with an accelerative equation in terms of only the iterandm x t . For achieving this, we recall Eq. (56) and obtain

m orr x t

N t X

x t0 z0 jm x t h i

N t0 j XH m x t

m1 x t h i UIIf the adjoint cyclic boundary condition N

t

X

N

t0 z0

1 is satised,the scheme reduces tom orr x t

m x t N

t0 j XH m1 x t

m1 x t h i UPAs argued before, we should realize that scheme (72) is exactly valid only if the

adjoint cyclic boundary condition expressed by Eqs. (59) and (61) is satised exactly.If for any reason one does not want to invest that much time in solving Eq. (61) withvery high accuracy, one should simply implement

m orr x t

m x t N

t0 j XH m x t

m1 x t h i" j m x t h i UQ

with " the residual operator that is zero only when the adjoint cyclic boundarycondition N t X

N t0 z0 1 is satised exactly:" N t

X N t0

z0

1 UR

As long as " is small enough, this accelerative scheme should, when applied beforeeach standard reload operation, result in an accelerated convergence towards the



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exact perturbation in the equilibrium cycle EOC nuclide density distribution,x t

. We would like to point out that, in principle, schemes (7) and (72) inde-

pendently contribute to the acceleration of the total numerical strategy for obtainingthe perturbed constrained equilibrium cycle solutions as functions of the reloadoperator and feed enrichment perturbations X 3 X X and x p 3 x p x p .So if scheme (7) an acceleration a 1 and scheme (72) an acceleration a 2, the totalacceleration e ect is quantied by the product 1 2. Further, we point out thatin our numerical study we did not implement any additional classical numericalacceleration strategy, such as for example Gauss-Seidel (GS) acceleration. If onewould do so, such an additional numerical acceleration strategy should simply addan extra factor to the formula for a, for example GS 1 2. For the GPT-basedschemes, the more favourable convergence properties arise from their inherentlymore favourable mathematical structure in the case of modest perturbations. It isinteresting to point out that the dependences of a 1 and a 2 on d X are quite di erent.From scheme (23) it is quite obvious that the smaller X (and thus the L i and F i),the faster the convergence of scheme (7). Now, in order to gain more knowledgeabout the inuence of X on the equilibrium cycle convergence behaviour of theGECDPT (level 2) acceleration, we recall Eq. (56), which expresses the perturbedreload operation in terms of the nuclide density perturbations for the (m+1) th andthe m th iterand,

m 1 x t 0

X x t

x p

XH m x t

x p

x p ; US

m x t 0 X x t

x p XH m1 x t x p x p UT

and subtract the latter from the former to obtain

2 m 1 x t 0 XH 2 m x t UU

with @2 m 1 x t 0 dened as2 m 1 x t 0

m 1

x t 0 m

x t 0 UVDuring the depletion that follows the reload operation, the 2 m 1 x t 0 is even-tually translated into 2 m 1 x t via numerical solution of the burnup equations.Obviously, since a depletion operator will always have a norm which is smaller thanunity, 2 m 1 x t will have a smaller norm than

2 m 1 x t 0 :2 m 1 x t 2 m x t 0

de letion < 1 UW

We note that, using denition (78) and the perturbed adjoint reload operator XH,the level 2 accelerative scheme can be rewritten as



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2 m orr x t

XHN t0 j

2 m x t h i; VHwhich implies that

1

2

2 m 1 orr x t

2 m x t 0 XHN t0 VI

Realizing that XHN t0 can be decomposed into its xed component XN t0 and its variable component XN t0 , it becomes clear that the dependence of a 2 onX is less pronounced than the dependence of a 1 on X and of a somewhat more

complex nature. From the numerical results that are presented at the end of thispaper it can be concluded that, for most (magnitudes of) choices for X, 1a2 remainssignicantly below unity (thus realizing a speed-up) and that the maximum speed-up(which is inversely proportional to XHN t0 occurs for X 0.6. Parallel numerical solution of the adjoint elds N t

The practical utility of the adjoint equations, like that of the forward quasi-staticequations, depends on how easily they can be solved numerically. Whereas the

acquisition of the adjoint elds i and

C i is a rather straightforward task from anumerical point of view, the numerical determination of N t dictates a somewhat

more involved and computationally more demanding procedure. The only di cultyin solving the i and the C

i from the inhomogeneous equations (11) and (41) arises

from the property of the operator L i l Fi that it is singular in the sense that afundamental solution (the fundamental adjoint ux i ) exists for the homogeneous

equation

L i l Fi

i 0 VP

due to which this fundamental solution i must be ltered out after each iterative

step for determining the i and the Ci , as described in Eq. (17). Due to the singu-

larity property of L i l iFi , the iterative solution of Eqs. (11) and (41) will featurea relatively slow convergence rate. As far as the acquisition of the N t is concerned,

the calculational ow is very similar to that for the forward equilibrium cycle itera-tion, except that the adjoint burnup calculations proceed backward in time. As isshown in Eq. (42), the ux adjoint source Q i at t i depends on an integral of N

t over the future time interval t i; t i 1 . Further, the nal value of N t at the end of each time interval is xed by the discontinuity condition (48). Its magnitude dependsnot only on the future behaviour of N t but also on the C i and P

i dened at the

nal time of the interval. Thus, the adjoint equations have to be solved backward intime. For solving the adjoint limit cycle iterations for determining N t , the owchart depicted in Fig. 2 can be implemented.



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The inner loop depicted in Fig. 2 also depicts the computational ow chart foreach adjoint burnup calculation from y t to y t 0 . We note here that theN t can be thought of as the collection of fully decoupled individual adjointnuclide density elds x q t corresponding to the response quantities x q t

that is,the equilibrium cycle EOC nuclide density of nuclide type q in node J):N t x q t Y 1; F F F; x ; q 1; F F F; nnn o VQ

The parallellisation of the calculation of N *(t) simply consists of distributing thecalculation of the di erent adjoint eld collections x q t over di erent parallelprocessors. An example of this is illustrated in Fig. 3.

Fig. 2. Backward ow chart for iterative calculation of N t0 such that the adjoint cyclic boundarycondition (61) is satised.



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We stress that, in this study of methodology development, we allowed ourselvesthe simplication of working with only 3 nuclide types: 238 U, 235 U and 239 Pu and nossion products taken into consideration. Nevertheless in principle, the DPT form-alism presented here allows any number of nuclide types (ssile nuclides, ssionproducts, absorber nuclides, etc.) to be taken into consideration. We indicate thatthe size of the largest adjoint object to be stored, N t0 is proportional to thesquare of the number of nodal nodes in the system and to the square of the numberof nuclide types taken into consideration. With present-day storage facilities, thisshould be no practical problem for most cases concerning realistic sizes of coresfeaturing octant symmetry. Further, the computational e ort involved in obtaining

N

t0 is only linearly proportional to the number of nodal nodes in the system andto the number of nuclide types taken into consideration. For two-dimensional nodalcore models to be used in reload pattern design, this should be manageable. Sincethe correction scheme (72) is only `pushing' the perturbed equilibrium limit cycleiteration in the right direction, one could even simply choose to calculate only thepart of N t0 associated with the nuclides that play the most dominant role indetermining the ux distributions and in the burnup process. The remaining part of N t0 then consists of zero elements and hence does not play a role in the correc-tion scheme (72). Then, although the correction e ected by (72) directly a ects onlythe most important nuclides and does not directly a ect the less dominant nuclides,

its e ect on the less dominant nuclides will be rather quickly conveyed through thetransmutation scheme in which the dominant nuclides are generally positioned atthe roots.

Fig. 3. Parallellisation of the solution of N t over N processors; the rst index indicates the responsenode, the second the nuclide type.



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7. Example of an application: acceleration of equilibrium cycle feed enrichmentminimization for a PWR core

The formalism described here has been programmed in MPI-Fortran and imple-mented on the massively parallel computer available at Delft University of Technology.It has been applied for acceleration of a cyclic interchange optimization procedure, inwhich a (local) optimum for a prespecied objective function is found by stepwiseimprovement through the evaluation of all possible octant-symmetric ternary (invol-ving three elements) and binary (pairwise, so involving two elements) interchanges of elements in the reference fuel element trajectory representation H . The optimization casefor the test 4-batch refueled PWR core containing 384 fuel elements considered was

. Minimization of the required feed enrichment (FE) for the constrained equili-

brium cycle, subject to the conditions that the radial power peaking factor f should remain below 1.8 and that the reloading should occur in an octant-symmetric way

With constrained equilibrium cycle we mean that the feed enrichment (symbolizedhere simply by the fresh fuel composition N F )is dictated by the condition that the l -eigenvalue at EOC should exactly equal 1. Or, in other words, the EOC-e ectivemultiplication factor l 1 the uncontrolled core (without reactivity control, such ascontrol rods or soluble boron) is constrained to be 1. The overall assumption is thatthe cycle length and the power level are constant. The feed enrichment satisfying this

condition is referred to as the minimal required feed enrichment (MRFE). Should,for the specic reload pattern in question and for the chosen power level and cyclelength, the feed enrichment be chosen to be smaller than MRFE, the core wouldbecome subcritical before EOC and thus the envisaged operation cycle would not befeasible. Obviously, this MRFE is a function of the choice for the reload operator Xand thus of the reload pattern H. Since in the search procedure the MRFE needs tobe calculated for each candidate reload pattern H, several equilibrium limit cycleiterations (with the FE xed) were performed within the loop in which MRFE isdetermined. In this picture, the limit cycle iteration can be regarded as another typeof inner iteration. The geometry of the test PWR core is illustrated in Fig. 4, in

which also the reloading structure of the starting reload pattern in the search pro-cedure (chosen intuitively to realize a low power peaking factor for satisfying theconstraint f 4 1.8) is indicated.

Naturally, since the cyclic interchange optimization algorithm is inherently paral-lel, the search procedure has been parallellized too in the sense that the workload of evaluating all the interchanges (pattern perturbations) is uniformly distributedamong the di erent available processors. In the two-dimensional 1 12-group nodalcore model (van Geemert, 1999) adopted in this study, the radial power peakingfactor is dened as

f m xtot l

; 1; F F F; x& ' VR



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with the nodal power in node I and N the total number of nodes. In the MRFEcalculation, iterations have taken place at three di erent levels. The two innermostiteration levels are depicted in the forward ow chart shown in Fig. 5 representingthe equilibrium cycle iteration for a specic xed N F .

We now dene and specify the following convergence parameters and criteriaeadopted in our study. For monitoring the convergence of scheme (1) or (7) in Fig. 5,the convergence parameter " i 1 for iterative step i1 was dened as

" i 1 m j ; i1 ; i11

; i11j; 1; F F F; x& ' VS

For monitoring the convergence of the equilibrium limit cycle iterations, the con-vergence parameter " i 2 EC for iterative step i 2 was dened as

" i 2 EC m jx i2 q ; t0 x

i21 q ; t0 x i21 q ; t0

j; 1; F F F; x ; q 1; F F F; nn@ A; VTand for monitoring the convergence of the FE iterations, the convergence parameter" i 3 p for iterative step i 3 was dened as

Fig. 4. Octant-symmetric test PWR core. The di erent shades of grey indicate the di erent fuel ages: thedarker, the higher the burnup.



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"i 3

p m j

l 1 i3 l 1 i3 1 l 1 i3 1 j; j1 l

1

i3 j@ A VUThe following convergence criteria were applied for evaluating both the reference

and the perturbed reload pattern:

" 4 106 ; " 4 105 ; " p 4 105 VV

For solving the adjoint eld equations, similar convergence parameters and cri-

teria were adopted as for the forward equations.In order to establish "p 4 105, generally about 11 or 12 outer iterations were

required, using the outer iterative scheme

Fig. 5. Forward ow chart for the iterative calculation of x t such that the cyclic boundary condition issatised and thus the equilibrium cycled is reached, for a specic xed N F .



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p i p i11 l 1 i 1

l 1 i 1 l 1 i 2 p i1 p i2 VW

Some examples have been isolated of reload operator perturbations of di erentmagnitudes, involving binary, ternary, 6-fold and 10-fold interchanges of elementsin the reference fuel element trajectory representation H . For these examples and inthe entire optimization procedure, we have calculated the perturbed constrainedequilibrium cycle solutions using the formalism described in this paper, so byapplying the system (7)

m 1 m j L H l m FH m h i a 1 j m h i 1

l m 1 l m j L H l m FH

m

h i j FH m h iVbbbbbbbbX

WH

in order to obtain lH; Hl l ; , instead of using the power method forsolving L l F 0 (Eq. (1)), and by applying the EOC correction

m orr x t

m x t N

t0 j XH m x t

m1 x t h iprior to each implementation of the reload operation that can be written in terms of nuclide density perturbations as

m 1 x t 0 X x t

x p XH x m orr t

x p x p WIIn Tables 1 and 2, the numbers of iterative steps required for convergence as

dened in (88) are listed for the di erent considered cases, consisting of reloadoperator perturbations of di erent magnitudes, involving binary, ternary, 6-fold and10-fold interchanges of elements in the reference octant-symmetric fuel element tra-

jectory representation H .For the large ctitious 4-batch refueled PWR core illustrated in Fig. 4 and starting

from the arbitrary reload pattern (chosen intuitively to realize a low power peakingfactor for satisfying the constraint f 4 1.8) that is illustrated in this picture, we havesearched for the loading scheme associated with the lowest MRFE subject to f 4 1.8$ and the EOC constraint l 1. For this, the ternary interchange optimizationprocedure was applied. The search results are illustrated graphically in Fig. 6 andFig. 7. The nal search result (that is, the reload pattern corresponding to the lowestencountered MRFE subject to f 4 1.8) is plotted in Fig. 8.

In Table 3, the improved MRFEs and their associated power peaking factors are

listed as a function of the number of adjoint eld adjustments (= number of for-ward MRFE iterations). Also indicated are the required cumulative processing timeson the parallel `Vermeer' CRAY-T3E, using 52 processors and applying the GPT-



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Table 1Average numbers of required `level 1' iterative steps for solving the ux eigenvalue equations for" 4 106 , with and without application of the single time step GPT method

Magnitude of permutation GPT-accelerated Non-accelerated

Binary 4 50Ternary 6 506-Fold 8 5010-Fold 10 50

Table 2Average numbers of required `level 2' iterative steps for achieving equilibrium cycle convergence"EC 4 105 , with and without application of the EOC-correction (72)Magnitude of permutation GECDPT-accelerated Non-accelerated

Zero 3 12Binary 4 12Ternary 4 126-Fold 5 1210-Fold 6 12

Fig. 6. Convergence towards the minimized MRFE as a function of the cumulative number of adjointeld adjustments.



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Fig. 7. Convergence towards the minimized MRFE.

Fig. 8. The octant-symmetric reload pattern corresponding to the minimized MRFE (subject to f 4 1:8).The darker the colour of the fuel assemblies, the higher their burnup.



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based accelerative techniques. The last two columns contain estimates of the cumu-lative CPU-times that would have been required in case no parallellization or/andno GPT-based accelerative techniques would have been applied. We note that a

certain fraction of the cumulative CPU time in the 4th column arises from thecomputational e orts invested in solving the adjoint eld equations.

8. Analysis of the achieved speed-up

In our sequential implementation for the core depicted in Fig. 4 (featuring 52response nodes in the octant), the solution of the i took about as much CPU-timeas one reference forward MRFE iteration, whereas the acquisition of N t0

took

about 5 times as much time, per response nuclide type. So obviously (and as expec-

ted), the computation of the adjoint elds certainly requires a non-trivial amount of computational e ort! However, for a design purpose such as reload pattern optimi-zation, this investment pays o extremely well in the case where there are very large

Table 3Listing of the search information

Cumulative numberof adjoint eldadjustments

MRFE(percent)

Powerpeakingfactor

CumulativeCPU-time (s)using GPT-basedacceleration and 52parallel processors

Estimate of cumulative CPU-time(s) using no GPT andno parallellization

1 1.5800 1.4579 740 648,1002 1.5690 1.6494 1480 1,296,2003 1.5613 1.7563 2220 1,944,3004 1.5570 1.7476 5 1.5537 1.7561 6 1.5515 1.7710 7 1.5499 1.6214 8 1.5483 1.6327 9 1.5469 1.6488 10 1.5458 1.7665 7400 6,481,00011 1.5445 1.7157 12 1.5426 1.7027 13 1.5421 1.6828 14 1.5408 1.7271 15 1.5404 1.7313 16 1.5401 1.7669 17 1.5400 1.7618 18 1.5398 1.7617

19 1.5395 1.7607 20 1.5394 1.7655 21 1.5393 1.7657 22 1.5392 1.7660 23 1.5391 1.7619 17,020 14,906,30024 1.5391 1.7621 17,760 15,554,400



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numbers of pattern changes to be evaluated. If these evaluations can be acceleratedconsiderably using the obtained adjoint elds (as indicated by the results in Tables 1 3), the gain in computational e ciency will be very signicant indeed. Suppose forexample that we have (nn) nuclide types to be included in N t0 . The CPU-timerequired for calculating N t0 is then about [5(nn)+1]T MRFE (T MRFE symbolizesthe average total time required for determining the minimally required feed enrich-ment associated with a certain reload pattern, without the use of perturbation theoryin any of the three iteration levels). However, if the average acceleration due to theuse of the adjoint elds is a 1a2 and a parallellization over Z processors is applied, the

gain G in computational e ciency for the case of evaluating M perturbed reloadpatterns can be expressed by

G

za11

a1M

1 Z ; onl level 1 eler tion

Za 1a2

1a1a2M

5 nn 1 Z level 1 2 eler tion

VbbbbbbbbX

WP

For very large M the limiting case lim M 3 I G Z ; M ; a1;2; nn

Za 1a2 will be

approached. Obviously, a fully sequential approach (no parallelization) correspondsto Z=1. In Figs. 9 and 10, the logarithm of G is plotted as a function of M for thecase a 1=10, a 2=2 and for di erent combinations of speed-up mechanisms applied.

Fig. 9. Speedup behaviour in case of three nuclides included in N t0 .



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For Figs. 9 and 10, the numbers of nuclides types accounted for in N t0 are 3 and40, respectively.In these pictures one can see that it is not always by denition advantageous to

calculate and apply N t0 in addition to calculating and applying the i . In the

case of only 3 nuclide types accounted for, this already o ers computationaladvantage for low values of M (starting from M=400), whereas in the case of thelarge number of 40 nuclide types it would not become advantageous until M exceeds4000. However we should recall our earlier remark that for speed-up purposes onedoes not necessarily have to include all of the present isotopes in N t0

, but only

the most dominant ones. In this case, less time has to be spent on calculating N

t0 at the cost of a limited loss of accelerative power.9. Conclusions

We conclude that the GPT-based accelerated iterative methods described in thispaper are well applicable to a heuristic equilibrium cycle optimization procedure thatis based on assessing the e ects of many di erent modest permutations in the loadingscheme. By this application, a considerable reduction of the computational costs for

the limit cycle iterations is realized. The price to be paid, the necessity of solving anextensive set of adjoint equations a few times during the search procedure, will gen-erally pay o easily if the evaluations of several thousands of candidate schemes can

Fig. 10. Speedup behaviour in case of 40 nuclides included in N t0

.



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be signicantly accelerated using the adjoint elds. Of interest here is that, for theseGPT-based schemes, the more favourable convergence properties arise from theinherently more favourable mathematical structure of the iterative equations in thecase of modest perturbations. When setting up the accelerated iterative expressionfor the equilibrium cycle, an adjoint reload operation invariance equation emergesfor the adjoint elds. Due to this, the acquisition of the adjoint time-dependentnuclide density distribution consists of an iterative procedure oriented backwards intime, featuring solution of adjoint burnup equations and application of adjoint reloadoperations until su cient convergence has been realized. Both the solution of theadjoint eld equations and the implementation of the applied heuristic optimizationalgorithm could be parallellized and executed on the parallel `Vermeer' CRAY-T3Eavailable at Delft University of Technology. From the speed-up that could be rea-lized, it can be concluded that the combination of parallellism and generalized per-turbation theory o ers the opportunity to perform exhaustive, fast and accuratesampling of the solution space for the equilibrium cycle optimization problem.

Acknowledgements

The authors would like to express special gratitude to the Delft Center for High-Performance Computing (HPAC) for the use of its `Vermeer' CRAY-T3E massivelyparallel computer.

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